On calibrated and separating sub-actions

Consider a transitive expanding dynamical system $\sigma: \Sigma \to \Sigma$, and a H\"older potential $A$. In ergodic optimization, one is interested in properties of $A$-maximizing probabilities. Assuming ergodicity, it is already known that the projection of the support of such probabilities is contained in the set of non-wandering points with respect to $A$, denoted by $\Omega(A)$. A separating sub-action is a sub-action such that the sub-cohomological equation becomes an identity just on $\Omega(A)$. For a fixed H\"older potential $A$, we prove not only that there exists H\"older separating sub-actions but in fact that they define a residual subset of the H\"older sub-actions. We use the existence of such separating sub-actions in an application for the case one has more than one maximizing probability. Suppose we have a finite number of distinct $A$-maximizing probabilities with ergodic property: $\hat \mu_j$, $j \in \{1, 2, ..., l\}$. Considering a calibrated sub-action $u$, under certain conditions, we will show that it can be written in the form $$u (\mathbf x)= u (\mathbf x^i) + h_A(\mathbf x^i, \mathbf x),$$ for all $\mathbf x \in \Sigma$, where $\mathbf x^i$ is a special point (in the projection of the support of a certain $\hat \mu_i$) and $h_A$ is the Peierls barrier associated to $A$.